Andrei Grekov (Stony Brook); November 21, 2024
In this talk, I will explore the relation between the generalized equivariant cohomology theories of the moduli space of instantons on \mathbb{C}^2 and the famous family of integrable systems: Calogero-Moser, Ruijsennars-Schneider, and DELL. We introduce the so-called \theta-transforms of the qq-characters vev’s, defined as integrals of certain classes in these cohomology theories, and relate them to quantum spectral curves of the corresponding integrable systems. The solution to the quantum spectral curve equation is constructed in a natural way as well. In the second half, I will explain the orbifolded version of all the notions defined above, which corresponds to the replacement of the moduli space of instantons with the affine Laumon space. Such treatment allows one to obtain the Lax matrices of the integrable systems in question in a new form, as well as the eigenvectors of these matrices. In the end, I will briefly explain how the above results help to rederive the quantum-classical duality between the trigonometric degenerations of the considered integrable systems and the corresponding spin chains, as well as shed some light on the spectral duality for them.